Let $X_n\xrightarrow[d]{}N(0,\sigma^2_x)$ and $Y_n\xrightarrow[d]{}N(0,\sigma^2_y)$.
$X_n, Y_n$ are not independent.
Can I say that $\left( \begin{array} {} X_n \\ Y_n \end{array} \right)\xrightarrow[d]{}N(\mathbf{0},\mathbf{C})$, with $\mathbf{C}$ a variance-covariance matrix?
Would $\mathbf{C}=\left( \begin{array}{ccc} \sigma^2_x & \lim Cov(X_n,Y_n) \\ \lim Cov(Y_n,X_n) & \sigma^2_y \end{array} \right)$ ?
The answer is no because if $X,Y$ are not independent, they can be jointly non-normal distributed.
That is because dependency implies a Copula function $c$. Only for the Gaussian Copula or Independence Copula, the joint distribution is Normal aswell:
$$f(x,y)=f(x)\cdot f(y)\cdot c\left(F^{-1}(x),F^{-1}(y)\right)$$
For example, using an asymmetric Copula, such as Gumbel Copula, would result in an asymmetric joint distribution, hence non-normal.